Only the inner point of the two lines are bound by the relation, not the outer points (or lengths).
I did some experimenting, and I'm seeing the same thing you showed. When applying a symmetric relation to lines it apparently only affects the location of the lines, not their lengths. I use the Mirror function extensively in sketches (often enough that I have a Mouse Gesture for it), and rarely apply the symmetric relation manually, so had never encountered the issue before. I'd suggest deleting the line (or lines) on one side, then mirror the others. That will produce the result that I believe you're looking for.
Actually, sometimes no endpoints gets bound, only a "colinear" type binding.
When dealing with lines in sketches it helps to treat the lines and their end points as separate entities, you're better off either making both of the end points symmetric to the respective partners, or making one symmetric and using an equal relation.
It's always been like this. Annoying.....
The way to have them be truly symmetric, draw your mid plane construction line and mirror. The symmetric relation only deals with the angle of the line(s) when adding it to some entities.
Symmetric has never been a go-to relation for me. Too fussy for most use, except in special circumstances. I would approach from 2 directions, depending on mood:
1: Draw half the pattern and mirror.
2: Draw full pattern, with center-line. Use Mid-point & Equal relations.
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